Double layer, capacitance Helmholtz

Figure 2.9, it can be seen that the interfacial capacitance does show a dependence on concentration, particularly at low concentrations. In addition, whilst there is some evidence of the expected step function away from the pzc, the capacitance is not independent of V. Finally, and most destructive, the Helmholtz model most certainly cannot explain the pronounced minimum in the plot at the pzc at low concentration. The first consequence of Figure 2.9 is that it is no longer correct to consider that differentiating the y vs. V plot twice with respect to V gives the absolute double layer capacitance CH where CH is independent of concentration and potential, and only depends on the radius of the solvated and/or unsolvated ion. This implies that the dy/dK (i.e. straight lines joined at the pzc. Thus, in practice, the experimentally obtained capacitance is (ddifferential capacitance. (The value quoted above of 0.05-0,5 Fm 2 for the double-layer was in terms of differential capacitance.) A particular value of (di M/d V) is obtained, and is valid, only at a particular electrolyte concentration and potential. This admits the experimentally observed dependence of the double layer capacity on V and concentration. All subsequent calculations thus use differential capacitances specific to a particular concentration and potential. 

Double-layer capacitance The capacitance C owing to the two Helmholtz layers at the electrode solution interface. 

If Nt rises above 1013cm 2, it may be anticipated that the complete emptying of such states will affect the potential drop in the Helmholtz layer. From above, the change in potential is rO = rNtfJNM, where Nm is the maximum possible number of surface states. Writing y = riVt/iVM, we may approximate y from the estimated double-layer capacitance of 10-20/iFcm-2 this gives y (0.08-0.16) x 10"13 Nt V. The rate constants k, and ka will also depend on yft specifically... 

The potential ([i is called the potential of the outer Helmholtz plane. The diffuse double layer starts at the outer Helmholtz plane, where the potential is ( ). It is this value of the potential, rather than (j), that must be used in Eqs. 14G and 15G, to relate the surface charge density and the diffuse-double-layer capacitance to potential. 

Tafel slopes for the anodic and the cathodic process double-layer capacitance ( lF/cm ) capacitance of the Helmholtz double layer capacitance of the diffuse double layer double-layer capacitance at 0 = 0 double-layer capacitance at 0 = 1 adsorption pseudocapacitance (llF/cm ) adsorption pseudocapacitance derived from the Langmuir isotherm... 

Figure 16. Schematic repmsentation of an electrochemical double layer at a mctal/elcctrolytc solution interface, (a) The jcllium double layer (with electron spill-over region contacts a layer of (ordered) solvent molecules, chemisorption of a negative ion is also shown, (b) Representation of the double layer capacitance as a series connection of the capacitance corresponding to the double layer of the metal surface, and the capacitance of (he Helmholtz layer al the solution side.

Figure 18. Estimation of the capacitance of the electrochemical double layer Cstiso as a function of the charge density, using a series connection of the capacitance of the double layer of the metal surface [Cm < 0) and the molecular Helmholtz layer (Csoi). It is found that the electrochemical double layer capacitance shows a maximum at around the point of zero charge, where the influence of the metal phase is strongest. This is in qualitative agreement with the experimental results (Fig. 17). The bars on the capacitance axis indicate 2 x 10 F/cm, the bars on the horizontal axis indicate 5 x 10 C/cm. ...

The double-layer capacitance is taken into account by assuming a simplified Helmholtz parallel plate model (1). On opening the circuit, the potential difference, V, across the double layer must be reduced by diminution of the charge on each plate. For a cathodic reaction, each electron being transferred from the metal to the solution side of the interface effects an elementary act of reaction and reduces the charge, q, on each plate. Consequently the rate of reduction of this charge is equal to the faradaic current, and Eq. (55) follows, y is assumed to differ from rj simply by the value of the reversible potential ... 

Double layer capacitance Excess charge in an electrode surface is compensated by a build-up of opposite-charged ions (Helmholtz layer), creating an electrical double layer. This layer is mathematically treated as a parallel plate capacitor. Typical values are on the order of tens of micro farads per cm. ... 

Double-layer capacitance Space-charge layer capacitance Capacitance of the Gouy-Chapman layer Capacitance of the Helmholtz layer Madelung constant Sauerbrey constant Distance... 

The EDLCs store charge electrostatically by using reversible adsorption of ions of the electrolyte on to high specific surface area materials, usually activated carbons. The charge separation occurs on polarisation at the electrode-electrolyte interface, which was first described by Helmholtz in 1853 as double layer capacitance. This is mathematically defined as ... 

As discussed in the previous chapter, the double-layer capacitance developed in electrochemical supercapacitors (ESs) is mainly due to net electrostatic charge accumulation and separation at the electrode-electrolyte interface. The net negative (or negative) charges such as electrons are accumulated near the electrode surface. At the same time, an equal number of positive charges such as cations are accumulated near the electrode surface at the electrolyte side, forming electric double-layers such as the Helmholtz and diffuse layers [1]. 

To date, it has been documented that ILs can be adsorbed onto various electrode surfaces. For example, Nanjundiah et al. found that several ILs used as electrolytes can induce double-layer capacitance phenomena on the surface of an Hg electrode and obtained the respective capacitance values for various ILs. Hyk and Stojek have also studied the IL thin layer on electrode surfaces and suggested that counterions substantially influence the distribution of IL. Kornyshev further discussed IL formations on electrode surfaces, suggesting that IL studies should be based on modern statistical mechanics of dense Coulomb systems or density-functional theory rather than classical electrochemical theories that hinge on a dilute-solution approximation. There are three conventional models that describe the charge distribution of an ion near a charged surface the Helmholtz model, the Gouy-Chapman model, and the Stern model. In the case of ILs, it remains controversial which model can best explain and lit the experimental data. 

The double-layer capacitance is composed of several contributions. In a geometrical sense the double layer in "supported" systems is represented by the compact "Helmholtz" or "Stem" layer. The electrostatically attracted solvated species reside in the "outer Helmholtz plane" (OHP), and specifically adsorbed species reside closer to the electrode in the "inner Helmholtz plane" (IHP). The double-layer structure is completed by a "diffuse" layer, composed of electrostatically attracted species at some distance from the electrode surface. The fuU thickness of the double layer can be defined as the external boundary of the diffuse layer separating it from the bulk solution, where the measured potential becomes equal to that of the bulk solution and no local potential gradient driven by the difference between the electrode potential ( )j and the solution potential can be determined (Figure 5-4). 

Total double-layer capacitance is composed of a series combination of the compact Helmholtz layer and the diffuse-layer capacitances as ... 

F/cm ) increases for higher applied voltages and higher concentrations of charged species C. For higher electrolyte concentrations (>0.1 M) DIFFUSE becomes constant, and total double layer capacitance is dominated by HELMHOLTZ Correction for the value of electrochemical potential ( )p at the point of separation between the Helmholtz and the diffuse layers is introduced as the second multiplier in Eq. 5-18 as ... 

Figure 6-1). Inside the double layer (the Helmholtz layer and to a lesser extent the diffuse layer) the charging of the interfaces occurs, resulting in the appearance of large double-layer capacitance... 

Helmholtz and diffuse-layer capacitances due to the presence of supporting electrolyte but not electroactive species discharged at e electrode. The bulk-resistance parameter due to migration remains the same in both representations. In the kinetic representation the double-layer capacitance (that is, the capacitance between the electrode and both supporting electrolyte and electroactive species in diffuse and Helmholtz layers) and the charge-transfer resistance (due to electroactive species in the compact Helmholtz layer) are replaced by the reaction capacitance in parallel with the reaction resistance... 

There is a similar quantity called the inner Helmholtz plane, where the potential is ( )j. This is determined by the radius of the anions in solution, which tend to be specifically adsorbed on the metal surface. A discussion of this phenomenon and its effect on the double-layer capacitance is outside the scope of this book. 

The traditional treatment of a double layer at electrode-electrolyte interfaces is based on its separation into two series contributions the compact ( Helmholtz ) layer and the diffusive ( dif ) layer, so that the inverse capacitance is... 

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